How to use image as plane with matcap turned on
The image that you want to keep is created by a material. The purpose of a matcap is to override all of the materials with the matcap material. This includes the material on your plane. As far as I know, there is no way to exclude an object from having the matcap material.

I would suggest using a custom matcap on your other objects instead, which is essentially just a material that looks like the matcap material. CG Masters did a fantastic tutorial on how to set that up here that also includes links to images that allow you to recreate the same matcap that you are using already.

Your other option is to enable viewport ambient occlusion. This won't make the objects look like they do with a matcap, but may help give you a bit more visual clarity while modelling by making corners and creases a bit darker. The option for that can be found just under where you enable matcaps. I would definitely suggest the first way though with the manual matcap material, as it's going to do exactly what you want.